Inner product space

Cauchy-Schwarz inequality

Let (𝑉,𝕂,⟨⋅|⋅⟩) be an inner product space. Then for any 𝑥,𝑦 ∈𝑉 we have the Cauchy-Schwarz inequality vec

|⟨𝑥|𝑦⟩|2≤‖𝑥‖2‖𝑦‖2⟨𝑥|𝑦⟩⟨𝑦|𝑥⟩≤⟨𝑥|𝑥⟩⟨𝑦|𝑦⟩

and equality holds iff 𝑥 and 𝑦 are linearly dependent.

Proof

proof

Particular examples

Probability theory

Using the Inner product space of real random variables we have

⟨𝑋,𝑌⟩2≤⟨𝑋,𝑋⟩⟨𝑌,𝑌⟩𝔼⁡[𝑋,𝑌]2≤𝔼⁡[𝑋,𝑋]𝔼⁡[𝑌,𝑌]

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